Heisenberg model (classical)
Encyclopedia
The Classical Heisenberg model is the 
case of the n-vector model, one of the models used in statistical physics
to model ferromagnetism
, and other phenomena.
, and a set of spins of the unit length
,
each one placed on a lattice node.
The model is defined through the following Hamiltonian
:
with
a coupling between spins.
case of the n-vector model, one of the models used in statistical physicsStatistical physics
Statistical physics is the branch of physics that uses methods of probability theory and statistics, and particularly the mathematical tools for dealing with large populations and approximations, in solving physical problems. It can describe a wide variety of fields with an inherently stochastic...
to model ferromagnetism
Ferromagnetism
Ferromagnetism is the basic mechanism by which certain materials form permanent magnets, or are attracted to magnets. In physics, several different types of magnetism are distinguished...
, and other phenomena.
Definition
It can be formulated as follows: take a d-dimensional latticeLattice (group)
In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which spans the real vector space Rn. Every lattice in Rn can be generated from a basis for the vector space by forming all linear combinations with integer coefficients...
, and a set of spins of the unit length
,each one placed on a lattice node.
The model is defined through the following Hamiltonian
Hamiltonian mechanics
Hamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton.It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without...
:
with
a coupling between spins.
Properties
- Polyakov has conjectured that, in dimension 2, as opposed to the classical XY model, there is no dipole phase for any
; i.e. at non-zero temperature the correlations cluster exponentially fast.
- The general mathematical formalism used to describe and solve the Heisenberg model and certain generalizations is developed in the article on the Potts modelPotts modelIn statistical mechanics, the Potts model, a generalization of the Ising model, is a model of interacting spins on a crystalline lattice. By studying the Potts model, one may gain insight into the behaviour of ferromagnets and certain other phenomena of solid state physics...
.
- In the continuum limit the Heisenberg model (2) gives the following equation of motion
-
-
- This equation is called the continuous classical Heisenberg ferromagnet equation or shortly Heisenberg modelHeisenberg modelThe Heisenberg model can refer to two models in statistical mechanics:*Heisenberg model , a classical nearest neighbour spin model*Heisenberg model , a model where the spins are treated quantum mechanically using Pauli matrices....
and is integrable in the soliton sense. It admits several integrable and nonintegrable generalizations like Landau-Lifshitz equation, Ishimori equationIshimori equationThe Ishimori equation is a partial differential equation proposed by the Japanese mathematician . Its interest is as the first example of a nonlinear spin-one field model in the plane that is integrable .-Equation:The IE has the form...
and so on.
See also
- Heisenberg model (quantum)Heisenberg model (quantum)The Heisenberg model is a statistical mechanical model used in the study of critical points and phase transitions of magnetic systems, in which the spin of the magnetic systems are treated quantum mechanically...
- Ising modelIsing modelThe Ising model is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables called spins that can be in one of two states . The spins are arranged in a graph , and each spin interacts with its nearest neighbors...
- Classical XY model
- MagnetismMagnetismMagnetism is a property of materials that respond at an atomic or subatomic level to an applied magnetic field. Ferromagnetism is the strongest and most familiar type of magnetism. It is responsible for the behavior of permanent magnets, which produce their own persistent magnetic fields, as well...
- FerromagnetismFerromagnetismFerromagnetism is the basic mechanism by which certain materials form permanent magnets, or are attracted to magnets. In physics, several different types of magnetism are distinguished...
- Landau-Lifshitz equation
- Ishimori equationIshimori equationThe Ishimori equation is a partial differential equation proposed by the Japanese mathematician . Its interest is as the first example of a nonlinear spin-one field model in the plane that is integrable .-Equation:The IE has the form...